91Ó°ÊÓ

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} \frac{\tan ^{-1} 3 x}{\sin ^{-1} x} $$

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow \infty} \frac{3 x}{\ln \left(100 x+e^{x}\right)} $$

Evaluate each improper integral or show that it diverges. \(\int_{-\infty}^{1} e^{4 x} d x\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow \pi / 2} \frac{3 \sec x+5}{\tan x} $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow-2} \frac{x^{2}+6 x+8}{x^{2}-3 x-10} $$

Evaluate each improper integral or show that it diverges. \(\int_{9}^{\infty} \frac{x d x}{\sqrt{1+x^{2}}}\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0^{+}} \frac{\ln \sin ^{2} x}{3 \ln \tan x} $$

Evaluate each improper integral or show that it diverges. \(\int_{1}^{\infty} \frac{d x}{\sqrt{\pi x}}\)

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} \frac{x^{3}-3 x^{2}+x}{x^{3}-2 x} $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 1^{-}} \frac{x^{2}-2 x+2}{x^{2}-1} $$